Paper · May 9, 2018

The Kähler-Ricci flow on manifolds with negative holomorphic curvature

Freid Tong

math.DGarXiv:1805.03562
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Abstract

We study the behaviour of the normalized Kähler-Ricci flow on complete Kähler manifolds of negative holomorphic sectional curvature. We show that the flow exists for all time and converges to a Kähler-Einstein metric of negative scalar curvature, recovering a result of Wu and Yau.

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Submitted
May 9, 2018
Subjects
math.DG
Ricci flowKähler–Ricci flowEinstein metricsKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (Ryosuke Nomura, 2016)2Nomura 2016 Cusp Kähler-Ricci flow on compact Kähler manifold (Jiawei Liu and Xi Zhang, 2017)4Liu 2017 The Kähler-Ricci flow on pseudoconvex domains (Huabin Ge, Aijin Lin and Liangming Shen, 2018)8Ge 2018 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)1Huang 2018 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)3Lee 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)7Guo 2020 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)6Jian 2023 On Kähler-Einstein Currents (Yifan Chen, Shih-Kai Chiu, Max Hallgren et al., 2025)5Chen 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

    Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same author Freid Tong · Shared terms “holomorphic sectional”, “negative scalar”, “sectional curvature”

  2. Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “negative holomorphic”, “holomorphic sectional”, “sectional curvature”

  3. Complete Kähler-Einstein metric on Stein manifolds with negative curvature

    Man-Chun Lee

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “complete Kähler”, “holomorphic sectional”, “normalized Kähler”

  4. Cusp Kähler-Ricci flow on compact Kähler manifold

    Jiawei Liu and Xi Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics

  5. On Kähler-Einstein Currents

    Yifan Chen, Shih-Kai Chiu, Max Hallgren et al.

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Same author Freid Tong

  6. estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation

    Wangjian Jian and Yalong Shi

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “normalized Kähler”

  7. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics

  8. The Kähler-Ricci flow on pseudoconvex domains

    Huabin Ge, Aijin Lin and Liangming Shen

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “complete Kähler”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.